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Solving Open Questions and Other Challenge Problems Using Proof Sketches
, 2001
"... . In this article, we describe a set of procedures and strategies for searching for proofs in logical systems based on the inference rule condensed detachment. The procedures and strategies rely on the derivation of proof sketchessequences of formulas that are used as hints to guide the search for ..."
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Cited by 28 (13 self)
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. In this article, we describe a set of procedures and strategies for searching for proofs in logical systems based on the inference rule condensed detachment. The procedures and strategies rely on the derivation of proof sketchessequences of formulas that are used as hints to guide the search for sound proofs. In the simplest case, a proof sketch consists of a subproofkey lemmas to prove, for exampleand the proof is completed by lling in the missing steps. In the more general case, a proof sketch consists of a sequence of formulas sucient to nd a proof, but it may include formulas that are not provable in the current theory. We nd that even in this more general case, proof sketches can provide valuable guidance in nding sound proofs. Proof sketches have been used successfully for numerous problems coming from a variety of problem areas. We have, for example, used proof sketches to nd several new twoaxiom systems for Boolean algebra using the Sheer stroke. Keywords: proof sk...
Automating the search for elegant proofs
 J. Automated Reasoning
"... The research reported in this article was spawned by a colleague’s request to find an elegant proof (of a theorem from Boolean algebra) to replace his proof consisting of 816 deduced steps. The request was met by finding a proof consisting of 100 deduced steps. The methodology used to obtain the far ..."
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Cited by 8 (4 self)
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The research reported in this article was spawned by a colleague’s request to find an elegant proof (of a theorem from Boolean algebra) to replace his proof consisting of 816 deduced steps. The request was met by finding a proof consisting of 100 deduced steps. The methodology used to obtain the far shorter proof is presented in detail through a sequence of experiments. Although clearly not an algorithm, the methodology is sufficiently general to enable its use for seeking elegant proofs regardless of the domain of study. In addition to (usually) being more elegant, shorter proofs often provide the needed path to constructing a more efficient circuit, a more effective algorithm, and the like. The methodology relies heavily on the assistance of McCune’s automated reasoning program OTTER. Of the aspects of proof elegance, the main focus here is on proof length, with brief attention paid to the type of term present, the number of variables required, and the complexity of deduced steps. The methodology is iterative, relying heavily on the use of three strategies: the resonance strategy, the hot list strategy, and McCune’s ratio strategy. These strategies, as well as other features on which the methodology relies, do exhibit tendencies that can be exploited in the search for shorter proofs and for other objectives. To provide some insight regarding the value of the methodology, I discuss its successful application to
The power of combining resonance with heat
 J. Automated Reasoning
, 1996
"... In this article, I present experimental evidence of the value of combining two strategies each of which has proved powerful in various contexts. The resonance strategy gives preference (for directing a program’s reasoning) to equations or formulas that have the same shape (ignoring variables) as one ..."
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Cited by 6 (5 self)
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In this article, I present experimental evidence of the value of combining two strategies each of which has proved powerful in various contexts. The resonance strategy gives preference (for directing a program’s reasoning) to equations or formulas that have the same shape (ignoring variables) as one of the patterns supplied by the researcher to be used as a resonator. The hot list strategy rearranges the order in which conclusions are drawn, the rearranging caused by immediately visiting and, depending on the value of the heat parameter, even immediately revisiting a set of input statements chosen by the researcher; the chosen statements are used to complete applications of inference rules rather than to initiate them. Combining these two strategies often enables an automated reasoning program to attack deep questions and hard problems with far more effectiveness than using either alone. The use of this combination in the context of cursory proof checking produced most unexpected and satisfying results, as I show here. I present the material (including commentary) in the spirit of excerpts from an experimenter’s notebook, thus meeting the frequent request to illustrate how a researcher can make wise choices from among the numerous options offered by McCune’s automated reasoning program OTTER. I include challenges and topics for research and, to aid the researcher, in the Appendix a sample input
Experiments concerning the Automated Search for Elegant Proofs
 Technical Memorandum ANL/MCSTM221, Mathematics and Computer Science Division, Argonne National Laboratory
, 1997
"... ..."
Logical Basis for the Automation of Reasoning: Case Studies 1
"... 2.1 Components of the clause language paradigm.................................... 4 ..."
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2.1 Components of the clause language paradigm.................................... 4
Logical Basis for the Automation of Reasoning: Case Studies
"... Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 The clause language paradigm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..."
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Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 The clause language paradigm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.1 Components of the clause language paradigm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 Interplay of the components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3 A fragment of the history of automated reasoning 19602002 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 4 Answering open questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 4.1 Robbins algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Preprint MCSP3871093 The Resonance Strategy*
"... Especially in mathematics and in logic, lemmas (basic truths) play a key role for proving theorems. In ring theory, for example, a useful lemma asserts that, for all elements x, the product in either order of 0 and x is 0; in twovalued sentential (or propositional) calculus, a useful lemma asserts ..."
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Especially in mathematics and in logic, lemmas (basic truths) play a key role for proving theorems. In ring theory, for example, a useful lemma asserts that, for all elements x, the product in either order of 0 and x is 0; in twovalued sentential (or propositional) calculus, a useful lemma asserts that, for all x, x implies x. Even in algorithm writing and in circuit design, lemmas play a key role: minus(minus(x)) = x in the former and NOT(AND(x,y)) = OR(NOT(x),NOT(y)) in the latter. Whether the object is to prove a theorem, write an algorithm, or design a circuit, and whether the assignment is given to a person or (preferably) to an automated reasoning program, the judicious use of lemmas often spells the difference between success and failure. In this article, we focus on what might be thought of as a generalization of the concept of lemma, namely, the concept of resonator, and on a strategy, the resonance strategy, that keys on resonators. For example, where in Boolean groups—those in which the square of every x is the identity element e—the lemmas yzyz = e and yyzz = e are such that neither generalizes the other, the resonator (formula schema) *** * = e, by using each occurrence of ‘‘star’ ’ to assert the presence of some variable, generalizes and captures (in a manner that is discussed in this article) both lemmas. Note that the cited resonator, if viewed as a lemma with star replaced by some chosen variable, captures neither cited lemma as an instance. Lemmas of a theory are provably ‘‘true’ ’ in the theory and, therefore, can be used to complete an assignment. In contrast, resonators, which capture collections of equations or collections of formulas that may or may not include truths, are used by the resonance strategy to direct the search for the information needed for assignment completion. In addition to discussing how one finds useful resonators, we detail various successes, in some of which the resonance strategy played a key role in obtaining a far better proof and in some of which the resonance strategy proved indispensable. The successes are taken from group theory, Robbins algebra, and various logic calculi.